I'm trying to parameterize a sphere through the stereographic projection.
This projection fails on the south pole, once I'm not working with $\infty$ as an element of the ring $\mathbb{R}$.
How can I define a map that sends me to the south pole? I'm struggling with this because all the points on the plane have already been taken.
Thank you.
Stereographic projection gives a homeomorphism $S^n\setminus \{p\}\to \Bbb{R}^n$, where $p$ is the point you project from. So, if you project from the north pole $N$, you won’t be able to define the map for $N$ (and for good reason - the sphere and Euclidean space are not homeomorphic). Anyway, you can use the another stereographic projection from the south pole $S$ to find a chart on $S^n$ containing $N$ if you need to and vice versa.